There are basically twelve volumes in the Yale Editions of the Private Papers of James Boswell (i.e., the Trade Edition; listed on Wikipedia here).** Of these it is—surprisingly—possible to download the text of six from the Internet Archive (IA). Surprisingly, because the Journals were published between 1950 and 1989 and I would have thought they were covered by copyright.
It is, however, quite difficult to find the six journals available on IA. The Journals appear under a variety of authors (James Boswell; Boswell; Belle de Zuylen; Pottle; and Doubleday and Company) and the eBooks are not listed on Open Library—which only lists (encrypted) DAISY editions—or LibraryThing. Indeed, I can find few links to any of the texts, though they have been downloaded many times.
And so I thought it might be useful to post a list of the journals here and update it as more etexts become available.
Boswell's London Journal, 1762–1763, ed. F.A. Pottle (1950). Not on IA, but searchable on Google Books here.
Boswell in Holland, 1763–1764, ed. F.A. Pottle (1952). On IA here.
Boswell on the Grand Tour: Germany and Switzerland, 1764, ed. F.A. Pottle (1953). On IA here.
Boswell on the Grand Tour: Italy, Corsica, and France, 1765–1766, ed. F. Brady and F.A. Pottle (1955). On IA here.
Boswell in Search of a Wife, 1766–1769, ed. F. Brady and F.A. Pottle (1956). On IA here.
Boswell for the Defence, 1769–1774, ed. W.K. Wimsatt Jr. and F.A. Pottle (1959). On IA here.
Boswell: The Ominous Years, 1774–1776, ed. C. Ryskamp and F.A. Pottle (1963). On IA here.
Boswell in Extremes, 1776–1778, ed. C.McC. Weis and F.A. Pottle (1970).
Boswell: Laird of Auchinleck, 1778–1782, ed. J.W. Reed and F.A. Pottle (1977).
Boswell: The Applause of the Jury, 1782–1785, ed. I.S. Lustig and F.A. Pottle (1981).
Boswell: The English Experiment, 1785–1789, ed. I.S. Lustig and F.A. Pottle (1986).
Boswell: The Great Biographer, 1789–1795, ed. M.K. Danziger and F. Brady (1989).
** I exclude Joshua Reynolds, Portraits, ed. F.W. Hilles (1952) and Boswell's Journal of a Tour to the Hebrides with Samuel Johnson, LL.D., 1773, ed. F.A. Pottle and C.H. Bennett (1961). Another ten volumes appeared in the Research Edition and there are three Independent Volumes (for a full list, see here).